Will Factoring solve the problem?

Algebra Level 3

$\frac{{\left(x+\dfrac{1}{x}\right)}^6-\left(x^6+\dfrac{1}{x^6}\right)-2}{{\left(x+\dfrac{1}{x}\right)}^3+\left(x^3+\dfrac{1}{x^3}\right)}$

If $x$ is a positive real number, then find the minimum value of the expression above.